I. Althofer, G. Das, D. P. Dobkin, D. Joseph, and J. Soares. On sparse spanners of weighted graphs. Discrete Comput. Geom., 9(1):81--100, 1993. An early version appeared in SWAT'90, LNCS V. 447.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Spanners and Approximate TSP in Weighted Graphs with Forbidden Minors Michelangelo Grigni Papa Sissokho Abstract Given an edge weighted graph G with n vertices and no K r minor and a small positive constant , we show that a simple greedy algorithm [1] finds a spanning subgraph approximating all shortest path distances within a factor of 1 , and with total edge weight at most log r Delta log n) times the weight of a minimum spanning tree. This result implies a quasi polynomial time approximation scheme (QPTAS) for the traveling ....

....the TSP approximation scheme we want a spanner G in G with very low stretch factor (s = 1 for a small fixed 0) and bounded tree weight; the tree weight bound appears in the exponent of our running time (see Theorem 3. 3) We find G by the following greedy algorithm of Althofer et al. [1]. Here (e) is the distance in G between the end points of e, which is 1 when they are not connected: Span(G = V; E) s) where initially E ; for all edges e 2 E in non decreasing w order do if s Delta w(e) dG (e) then add e to E return G In the resulting G = ....

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I. Althofer, G. Das, D. P. Dobkin, D. Joseph, and J. Soares. On sparse spanners of weighted graphs. Discrete Comput. Geom., 9(1):81--100, 1993. An early version appeared in SWAT'90, LNCS V. 447.

....remarks A deficiency of our PTAS is its dependence on # = c(G) OPT(G) where OPT(G) is the minimum cost of a 2 ECSS in G. We know of no hardness result justifying this dependency, and indeed a very similar dependency in the context of the TSP was eliminated using a known spanner construction [1, 2], which safely prunes some heavy edges out of G. We conjecture that a similar approach works for the 2 ECSS problem. Definition 10 Given s 1 and a 2 EC graph G with non negative edge costs, a (2 EC, s) spanner of G is a 2 ECSS G # in G such that OPT(G # ) s OPT(G) The cost ratio of such ....

I. Althofer, G. Das, D. P. Dobkin, D. Joseph, and J. Soares. On sparse spanners of weighted graphs. Discrete Comput. Geom., 9(1):81--100, 1993. An early version appeared in SWAT'90, LNCS V. 447. 10

....to this line of work. In MPLS, setting up the initial stack may require more memory than conventional routing problems, but once the stack is set up, the memory needed by each router to just forward the packets is very small. There has also been lot of work on finding sparse spanners of graphs [1], 3] However, these results are interesting only when the graph is not sparse, whereas the problems we address in this paper are non trivial even for bounded degree graphs. Another different (but related) large corpus of work has studied the problem of distance labeling of graphs [26] 21] ....

Ingo Alth ofer, Gautam Das, David Dobkin, Deborah Joseph, and Jose Soares. On sparse spanners of weighted graphs. Discrete & Computational Geometry, 9(1):81--100, 1993.

....following deterministic sequential algorithm. Order the edges of G in some fashion: say each edge has a unique ID, and list the edges in the increasing order of their IDs. If the edges do not have such IDs, a simple distributed way of achieving this is to make each edge choose a random real in [0, 1] as its ID. Consider the edges one by one from this list and delete them if they are currently in a cycle of length less than 1 2 log n. Clearly, when an edge is deleted it is in a cycle, so its deletion cannot disconnect the graph. Thus, in the end we will be left with a connected graph of ....

....sets with low stretch The algorithms in the previous section relied on the observation that if a graph has no cycles of length less than 1 2 log n, then it must have at most 2n edges. From this observation, one can, in fact, infer the following stronger statement. Proposition 4 (see, e.g. [1]) For every graph G on n vertices, there is a subgraph H with V (H) V (G) and with a linear number of edges, such that for every two vertices v, w V (G) dH (v, w) 1 2 log n) Proof: Start from the empty graph H. Consider the edges of G in some order. If the edge closes a cycle of ....

I. Althofer, G. Das, D. P. Dobkin, D. Joseph, and J. Soares. On sparse spanners of weighted graphs. Discrete Comput. Geom., 9:81--100, 1993.

....graph is said to be a t spanner if, between any pair of nodes, the distance in the subgraph is at most t times longer than the distance in the original graph. The value of t is thus the stretch factor associated with the subgraph. We encode the full mesh by using a spanner graph. It is shown in [2] that, for an undirected graph, there exists a polynomially constructible (2t 1) Spanner such that the number of links on the spanner is smaller than MdM 1=t e, where M is the number of nodes. We use the algorithm proposed in [2] to construct a spanner graph. This algorithm makes use of an ....

....We encode the full mesh by using a spanner graph. It is shown in [2] that, for an undirected graph, there exists a polynomially constructible (2t 1) Spanner such that the number of links on the spanner is smaller than MdM 1=t e, where M is the number of nodes. We use the algorithm proposed in [2] to construct a spanner graph. This algorithm makes use of an extension of a minimum weight spanning tree algorithm. B. Bandwidth Aggregation The process of bandwidth aggregation is similar to that of delay aggregation. There are two steps involved namely, full mesh generation and spanning tree ....

I. Althofer et al, "On Sparse Spanners of Weighted Graphs," Discrete & Computational Geometry, Vol. 9, pp. 81-100, 1993.

....just forwarding the packet, the total memory required is O(n) Furthermore, many previous results giving small storage allow the vertices to be labeled by the algorithm, whereas we make no assumptions on the vertex names. There has also been lot of work on finding sparse spanners of graphs [1, 3]. However, these results are interesting only when the graph is not sparse, whereas the problems we address in this paper are non trivial even for bounded degree graphs. Another different (but related) large corpus of work has studied the problem of distance labeling of graphs [23, 17, 10] ....

I. Althofer, G. Das, D. Dobkin, D. Joseph, and J. Soares. On sparse spanners of weighted graphs. Discrete & Computational Geometry, 9(1):81--100, 1993.

....useful to know all of the distances between them. 7 Knowing the distance of one Tracer from each area would likely allow a sufficient distance approximation between hosts in Seattle and hosts in Boston (Figure 6) We now generalize the above observations by applying the spanner algorithm [44] to distance map construction. A spanner of a graph is a subgraph where the distance between any pair of nodes is at most times larger than the distance in the original graph [45, 46] Formally, given a graph, a spanner is a subgraph , 1 such ....

....that the minimum spanner (a spanner with the minimum number of edges) is an NP complete problem. However, asymptotically, the algorithm of Althofer et al. generates, from a graph , a spanners whose edge count is in the same order of magnitude as the optimal spanner [44]. Fig. 7 presents the spanner algorithm of Althofer et al. 44] It first sorts, in increasing order, all the edges 7 We recognize that geographical distance does not directly relate to network distance (though often the two are related) for instance because of multi point traffic exchange ....

[Article contains additional citation context not shown here]

I. Althofer, G. Das, D. Dopkin, D. Joseph, and J. Soares, "On sparse spanners of weighted graphs," Discrete and Computational Geometry, vol. 9, pp. 81 -- 100, 1993.

....very useful to know all of the distances between them. 7 Knowing the distance of one Tracer from each area would likely allow a sufficient distance approximation between hosts in Seattle and hosts in Boston (Fig. 6) We now generalize the above observations by applying the spanner algorithm [21] to distance map construction. A We recognize that geographical distance does not directly relate to network distance (though often the two are related) for instance because of multi point traffic exchange between global ISPs. We use geographical locations here to simplify the ....

....A We recognize that geographical distance does not directly relate to network distance (though often the two are related) for instance because of multi point traffic exchange between global ISPs. We use geographical locations here to simplify the discussion. 8 Algorithm 3 ( spanner [21]) 1. sort by cost in non decreasing order 2. 2 3 3. for each edge ; in do 4. if 9640 3 ; 3 ; 5. 3 ; # Fig. 7. The spanner algorithm. spanner of a graph is a subgraph where the distance between any pair ....

[Article contains additional citation context not shown here]

I. Althofer, G. Das, D. Dopkin, D. Joseph, and J. Soares, "On sparse spanners of weighted graphs," Discrete and Computational Geometry, vol. 9, pp. 81 -- 100, 1993.

....each other and APs (defined in Section 3.1) The resulting distance information are advertised to IDMaps clients. Clients of IDMaps, such as SONAR or 12 ISPA ISPB ISPB ISPA raw distance Seattle ISPA ISPB ISPB ISPA Boston Figure 6: Distance Measurement Reduction Algorithm 3 (t spanner [45]) 1. sort E by cost c in non decreasing order 2. G 0 (V; E 0 ) E 0 ; 3. for each edge (u; v) in E do 4. if (t c( u; v) dG 0 (u; v) 5. E 0 (u; v) E 0 Figure 7: The t spanner algorithm. HOPS servers, collect the advertised distance information and construct distance ....

....very useful to know all of the distances between them. 7 Knowing the distance of one Tracer from each area would likely allow a sufficient distance approximation between hosts in Seattle and hosts in Boston (Figure 6) We now generalize the above observations by applying the t spanner algorithm [45] to distance map construction. A t spanner of a graph is a subgraph where the distance between any pair of nodes is at most t times larger than the distance in the original graph [46, 47] Formally, given a graph, G(V; E) a t spanner is a subgraph G 0 (V; E 0 ) E 0 E such that d G 0 ....

[Article contains additional citation context not shown here]

I. Althofer, G. Das, D. Dopkin, D. Joseph, and J. Soares, "On sparse spanners of weighted graphs," Discrete and Computational Geometry, vol. 9, pp. 81 -- 100, 1993.

....the distance between p and q. If S is a set of points in R d for some constant d, and the metric is the Euclidean metric, then we call the graph G a Euclidean t spanner. The problem of constructing spanners has been investigated by many researchers. For general metric spaces, Althofer et al. [1], and Chandra et al. 5] showed that a natural greedy algorithm computes, for any constant t 1, a t spanner with O(n 1 2= t Gamma1) edges, in O(n 3 4= t Gamma1) time. For the Euclidean case in R 2 , Keil and Gutwin [8] showed that for any constant t 1, a t spanner for S having O(n) ....

I. Althofer, G. Das, D. P. Dobkin, D. Joseph, and J. Soares. On sparse spanners of weighted graphs. Discrete Comput. Geom., 9:81--100, 1993.

....n leaves. The tree depth is at most the maximum number of iterations, considered above. Therefore there is a polynomial size collection of vertex sets in G, such that for any weighing w, one of them is a separator satisfying the conditions of Theorem 1(b) 4 The Span Algorithm Althofer et al. [2] introduced the following greedy algorithm to find an s spanner in an edge weighted graph G. The parameter s is at least one, and dG 0 (e) denotes the length of the shortest path in G 0 connecting the endpoints of edge e (the length may be 1) Span(G = V; E) s) G 0 (V; for all e ....

I. Althofer, G. Das, D. P. Dobkin, D. Joseph, and J. Soares. On sparse spanners of weighted graphs. Discrete Comput. Geom., 9(1):81--100, 1993. An early version appeared in SWAT'90, LNCS V. 447.

.... analysis) Graham and Winkler [GW85] study embeddings in ZZ d (from a graph theoretic motivation) Algorithmic applications in distributed computation and graph algorithms, have led to the notion of graph spanners, introduced by Peleg and Ullman [PU89] and later studied in many papers including [PS89, ADDJS90, CDNS92], and to low distortion embeddings in low dimensional real normed spaces by Linial, London and Rabinovich [LLR94] We extend this idea to deal with randomized algorithms by defining a probabilistic approximation of metric spaces by a set of simpler metric spaces S. Given a metric space M over ....

I. Althofer, G. Das, D. Dobkin, D. Joseph, and J. Soares. On Sparse Spanners of Weighted Graphs. In Discrete and Computational Geometry.

.... between the border nodes of the original graph (as explained later, this losslessness does not hold under multiple QoS parameters) The second step involves mapping the full mesh into a more compact topology, such as symmetricnode (simple node) star [15] minimum spanning tree [14] and t spanner [2]. Graph reduction is performed by pruning several links of the full mesh. The compact topology is then represented as a complex node, which is broadcasted to the rest of the network. One problem in the above approach is that the amount of lossyness that results from graph reduction is not known ....

I. Althofer et al. On sparse spanners of weighted graphs. Discrete & Computational Geometry, 9(1):81--100, 1993.

.... fdist G 0 (u; v) dist G (u; v)g: The subgraph G 0 is said to be a spanner for G if Stretch(G 0 ) Spanners have been studied in a number of papers, motivated by applications in diverse contexts such as distributed systems, communication networks, robotics and computational geometry (cf. [PU89, PS89, ADDJS93, ABP91, CDNS92]) The spanners we use for our present purposes are the low stretch, sparse, light weight spanners constructed by the simple greedy algorithm of [ADDJS93, CDNS92] Alternatively, one can employ the spanner construction algorithm of [ABP91] In case the variance in link prices in the matrix P is ....

.... in diverse contexts such as distributed systems, communication networks, robotics and computational geometry (cf. PU89, PS89, ADDJS93, ABP91, CDNS92] The spanners we use for our present purposes are the low stretch, sparse, light weight spanners constructed by the simple greedy algorithm of [ADDJS93, CDNS92]. Alternatively, one can employ the spanner construction algorithm of [ABP91] In case the variance in link prices in the matrix P is not extremely large (specifically, as long as the ratio between the largest and smallest price is bounded by a polynomial in n) that algorithm yields comparable ....

[Article contains additional citation context not shown here]

I. Althofer, G. Das, D. Dobkin D. Joseph and J. Soares. On sparse spanners of weighted graphs. In Discrete and Computat. Geometry, 9:81--100, 1993.

.... the border nodes of the original graph (as explained later, this losslessness does not hold under multiple QoS parameters) The second step involves mapping the full mesh into a more compact topology, such as symmetric node (simple node) star [20] minimum spanning tree [19] and t spanner [2, 15]. Graph reduction is performed by pruning several links of the full mesh. The compact topology is 3 then represented as a complex node, which is broadcasted to the rest of the network. One problem in the above approach is that the amount of lossyness that results from graph reduction is not ....

I. Althofer et al. On sparse spanners of weighted graphs. Discrete & Computational Geometry, 9(1):81--100, 1993. 24

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I. Althofer, G. Das, D. P. Dobkin, D. Joseph, and J. Soares. On sparse spanners of weighted graphs. Discrete Comput. Geom., 9(1):81--100, 1993. An early version appeared in SWAT'90, LNCS V. 447.

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I. Althofer, G. Das, D. Dobkin, D. Joseph, and J. Soares. On sparse spanners of weighted graphs. Discrete & Computational Geometry, 9:81--100, 1993.

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I. Althofer, G. Das, D. Dobkin, D. Joseph, and J. Soares. On sparse spanners of weighted graphs. Discrete & Computational Geometry, 9:81--100, 1993.

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I. Althofer, G. Das, D. Dobkin, D. Joseph, and J. Soares. On sparse spanners of weighted graphs. Discrete & Computational Geometry, 9:81--100, 1993.

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I. Althofer, G. Das, D. Dobkin, D. Joseph, and J. Soares. On sparse spanners of weighted graphs. Discrete & Computational Geometry, 9:81--100, 1993.

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I. Althofer, G. Das, D. Dobkin, D. Joseph, and J. Soares. On sparse spanners of weighted graphs. Discrete & Computational Geometry, 9:81--100, 1993.

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I. Althofer, G. Das, D. Dobkin, D. Joseph, and J. Soares. On sparse spanners of weighted graphs. Discrete & Computational Geometry, 9:81--100, 1993.

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I. Althofer, G. Das, D. Dobkin, D. Joseph, and J. Soares. On sparse spanners of weighted graphs. Discrete & Computational Geometry, 9:81--100, 1993.

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Ingo Althofer, Gautam Das, David Dobkin, Deborah Joseph, and Jose Soares. On sparse spanners of weighted graphs. Discrete & Computational Geometry, 9(1):81-100, 1993.

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I. Althofer, G. Das, D. Dobkin, D. Joseph, L. Soares, "On sparse spanners of weighted graphs", Journal of Discrete and Computational Computational Geometry, Vol. 9, no. 1, pp. 81-100, 1993.

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